论文标题
通过罗马总策略来主导两个图的直接产物
Dominating the direct product of two graphs through total Roman strategies
论文作者
论文摘要
给定一个没有孤立顶点的图形$ g $，$ g $的总罗马主导函数是一个函数$ f：v（g）\ rightarrow \ {0,1,1,2 \} $，使得每个带有标签0的顶点与带有标签2的顶点相邻，并且具有标签2的顶点，并且具有正面标签的阳性标签集，该标签的集合均与一个最小图的高度图表。 $ g $的总罗马统治号码$γ_{tr}（g）$是$ \ sum_ {v \ in V（g）} f（v）f（v）$的最小值值。在这项工作中，研究了直接产品$ g \ times h $的直接产品$ g \ times h $的总数。具体而言，给出了几种以上和下限的形状的关系，在$γ_{tr}（g \ times h）$和某些因子的经典统治参数之间。提出了直接产品图$ g \ times h $以$γ_{tr}（g \ times h）$实现较小的值（$ \ le 7 $）$，并提供$γ_{tr}（g \ times h）$的精确值，同时考虑各种特定的直接产品类别。
Given a graph $G$ without isolated vertices, a total Roman dominating function for $G$ is a function $f : V(G)\rightarrow \{0,1,2\}$ such that every vertex with label 0 is adjacent to a vertex with label 2, and the set of vertices with positive labels induces a graph of minimum degree at least one. The total Roman domination number $γ_{tR}(G)$ of $G$ is the smallest possible value of $\sum_{v\in V(G)}f(v)$ among all total Roman dominating functions $f$. The total Roman domination number of the direct product $G\times H$ of the graphs $G$ and $H$ is studied in this work. Specifically, several relationships, in the shape of upper and lower bounds, between $γ_{tR}(G\times H)$ and some classical domination parameters for the factors are given. Characterizations of the direct product graphs $G\times H$ achieving small values ($\le 7$) for $γ_{tR}(G\times H)$ are presented, and exact values for $γ_{tR}(G\times H)$ are deduced, while considering various specific direct product classes.
